Abstract

The T-matrix framework offers accurate and efficient modelling of electromagnetic scattering by nonspherical particles in a wide variety of applications ranging from nano-optics to atmospheric science. Its analytical setting, in contrast to purely numerical methods, also provides a fertile ground for further theoretical developments. Perhaps the main purported limitation of the method, when extended to systems of multiple particles, is the often-stated requirement that the smallest circumscribed spheres of neighbouring scatterers not overlap. We consider here such a scenario with two adjacent spheroids whose aspect ratio we vary to control the overlap of the smallest circumscribed spheres, and compute far-field cross-sections and near-field intensities using the superposition T-matrix method. The results correctly converge far beyond the no-overlap condition, and although numerical instabilities appear for the most extreme cases of overlap, requiring high multipole orders, convergence can still be obtained by switching to quadruple precision. Local fields converge wherever the Rayleigh hypothesis is valid for each single scatterer and, remarkably, even in parts of the overlap region. Our results are validated against finite-element calculations, and the agreement demonstrates that the superposition T-matrix method is more robust and broadly applicable than generally assumed.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
  36. J. Grand and E. C. Le Ru, “Practical implementation of accurate finite-element calculations for electromagnetic scattering by nanoparticles,” Plasmonics (2019).
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    [Crossref]
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    [Crossref]
  39. E. C. Le Ru and P. G. Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy: and related plasmonic effects (Elsevier, 2009).
  40. W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Severe loss of precision in calculations of T-matrix integrals,” J. Quant. Spectrosc. Radiat. Transfer 113(7), 524–535 (2012).
    [Crossref]
  41. W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “A new numerically stable implementation of the T-matrix method for electromagnetic scattering by spheroidal particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 153–168 (2013).
    [Crossref]
  42. D. Amos, “Algorithm 644, a portable package for Bessel functions of a complex argument and nonnegative order,” ACM Trans. Math. Softw. 12(3), 265–273 (1986).
    [Crossref]
  43. W. Somerville, B. Auguié, and E. Le Ru, “SMARTIES: User-friendly codes for fast and accurate calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 174, 39–55 (2016).
    [Crossref]
  44. H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
    [Crossref]
  45. D. Maystre and M. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26(9), 2201–2204 (1985).
    [Crossref]
  46. M. R. A. Majić, B. Auguié, and E. C. Le Ru, “Spheroidal harmonic expansions for the solution of Laplace’s equation for a point source near a sphere,” Phys. Rev. E 95(3), 033307 (2017).
    [Crossref]

2019 (1)

T. Martin, “T-matrix method for closely adjacent obstacles,” J. Quant. Spectrosc. Radiat. Transfer 234, 40–46 (2019).
[Crossref]

2017 (3)

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

M. R. A. Majić, B. Auguié, and E. C. Le Ru, “Spheroidal harmonic expansions for the solution of Laplace’s equation for a point source near a sphere,” Phys. Rev. E 95(3), 033307 (2017).
[Crossref]

J. Markkanen and A. J. Yuffa, “Fast superposition T-matrix solution for clusters with arbitrarily-shaped constituent particles,” J. Quant. Spectrosc. Radiat. Transfer 189, 181–188 (2017).
[Crossref]

2016 (2)

W. Somerville, B. Auguié, and E. Le Ru, “SMARTIES: User-friendly codes for fast and accurate calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 174, 39–55 (2016).
[Crossref]

B. Auguié, W. R. C. Somerville, S. Roache, and E. C. Le Ru, “Numerical investigation of the Rayleigh hypothesis for electromagnetic scattering by a particle,” J. Opt. 18(7), 075007 (2016).
[Crossref]

2015 (2)

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Accurate and convergent T-matrix calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 160, 29–35 (2015).
[Crossref]

2014 (2)

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 138, 17–35 (2014).
[Crossref]

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

2013 (2)

N. G. Khlebtsov, “T-matrix method in plasmonics: An overview,” J. Quant. Spectrosc. Radiat. Transfer 123, 184–217 (2013).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “A new numerically stable implementation of the T-matrix method for electromagnetic scattering by spheroidal particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 153–168 (2013).
[Crossref]

2012 (2)

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Severe loss of precision in calculations of T-matrix integrals,” J. Quant. Spectrosc. Radiat. Transfer 113(7), 524–535 (2012).
[Crossref]

2011 (1)

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58(5-6), 528–544 (2011).
[Crossref]

2008 (2)

L. Liu, M. I. Mishchenko, and W. P. Arnott, “A study of radiative properties of fractal soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109(15), 2656–2663 (2008).
[Crossref]

B. Stout, J. C. Auger, and A. Devilez, “Recursive T matrix algorithm for resonant multiple scattering: applications to localized plasmon excitations,” J. Opt. Soc. Am. A 25(10), 2549–2557 (2008).
[Crossref]

2004 (1)

Z. Zhong, S. Patskovskyy, P. Bouvrette, J. H. T. Luong, and A. Gedanken, “The surface chemistry of au colloids and their interactions with functional amino acids,” J. Phys. Chem. B 108(13), 4046–4052 (2004).
[Crossref]

2003 (1)

2002 (2)

V. G. Farafonov, “Applicability of the T-matrix method and its modifications,” Opt. Spectrosc. 92(5), 748–760 (2002).
[Crossref]

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple-scattering problems,” J. Mod. Opt. 49(13), 2129–2152 (2002).
[Crossref]

2001 (1)

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

1999 (1)

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
[Crossref]

1996 (3)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transfer 55(5), 535–575 (1996).
[Crossref]

Y.-Q. Jin and X. Huang, “Numerical T-matrix solution for polarized scattering from a cluster of spatially oriented, nonspherical scatterers,” Q. Appl. Math. 12(3), 154–158 (1996).
[Crossref]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996).
[Crossref]

1994 (1)

1991 (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London, Ser. A 433(1889), 599–614 (1991).
[Crossref]

1986 (1)

D. Amos, “Algorithm 644, a portable package for Bessel functions of a complex argument and nonnegative order,” ACM Trans. Math. Softw. 12(3), 265–273 (1986).
[Crossref]

1985 (1)

D. Maystre and M. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26(9), 2201–2204 (1985).
[Crossref]

1975 (1)

1973 (1)

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8(10), 3661–3678 (1973).
[Crossref]

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3(4), 825–839 (1971).
[Crossref]

1969 (2)

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45(6), 1417–1429 (1969).
[Crossref]

R. Millar, “Rayleigh hypothesis in scattering problems,” Electron. Lett. 5(17), 416–417 (1969).
[Crossref]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20(1), 33–40 (1962).
[Crossref]

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15–24 (1961).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Aiello, S.

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

Amos, D.

D. Amos, “Algorithm 644, a portable package for Bessel functions of a complex argument and nonnegative order,” ACM Trans. Math. Softw. 12(3), 265–273 (1986).
[Crossref]

Arnott, W. P.

L. Liu, M. I. Mishchenko, and W. P. Arnott, “A study of radiative properties of fractal soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109(15), 2656–2663 (2008).
[Crossref]

Auger, J. C.

Auger, J.-C.

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple-scattering problems,” J. Mod. Opt. 49(13), 2129–2152 (2002).
[Crossref]

Auguié, B.

M. R. A. Majić, B. Auguié, and E. C. Le Ru, “Spheroidal harmonic expansions for the solution of Laplace’s equation for a point source near a sphere,” Phys. Rev. E 95(3), 033307 (2017).
[Crossref]

W. Somerville, B. Auguié, and E. Le Ru, “SMARTIES: User-friendly codes for fast and accurate calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 174, 39–55 (2016).
[Crossref]

B. Auguié, W. R. C. Somerville, S. Roache, and E. C. Le Ru, “Numerical investigation of the Rayleigh hypothesis for electromagnetic scattering by a particle,” J. Opt. 18(7), 075007 (2016).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Accurate and convergent T-matrix calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 160, 29–35 (2015).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “A new numerically stable implementation of the T-matrix method for electromagnetic scattering by spheroidal particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 153–168 (2013).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Severe loss of precision in calculations of T-matrix integrals,” J. Quant. Spectrosc. Radiat. Transfer 113(7), 524–535 (2012).
[Crossref]

Barber, P.

Bi, L.

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 138, 17–35 (2014).
[Crossref]

Boreman, G. D.

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

Borghese, F.

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

Bouvrette, P.

Z. Zhong, S. Patskovskyy, P. Bouvrette, J. H. T. Luong, and A. Gedanken, “The surface chemistry of au colloids and their interactions with functional amino acids,” J. Phys. Chem. B 108(13), 4046–4052 (2004).
[Crossref]

Cadilhac, M.

D. Maystre and M. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26(9), 2201–2204 (1985).
[Crossref]

Cecchi-Pestellini, C.

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

Chew, W.

W. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press series on electromagnetic waves (Van Nostrand Reinhold, 1990).

Conny, J. M.

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20(1), 33–40 (1962).
[Crossref]

D’Archangel, J.

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

Denti, P.

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

Devilez, A.

Doicu, A.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Egel, A.

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

Eremin, Y.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Etchegoin, P. G.

E. C. Le Ru and P. G. Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy: and related plasmonic effects (Elsevier, 2009).

Farafonov, V. G.

V. G. Farafonov, “Applicability of the T-matrix method and its modifications,” Opt. Spectrosc. 92(5), 748–760 (2002).
[Crossref]

García de Abajo, F. J.

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
[Crossref]

Gedanken, A.

Z. Zhong, S. Patskovskyy, P. Bouvrette, J. H. T. Luong, and A. Gedanken, “The surface chemistry of au colloids and their interactions with functional amino acids,” J. Phys. Chem. B 108(13), 4046–4052 (2004).
[Crossref]

Gill, R.

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

Gomard, G.

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

Grand, J.

J. Grand and E. C. Le Ru, “Practical implementation of accurate finite-element calculations for electromagnetic scattering by nanoparticles,” Plasmonics (2019).

Heckenberg, N. R.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58(5-6), 528–544 (2011).
[Crossref]

Hodges, J. T.

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

Huang, X.

Y.-Q. Jin and X. Huang, “Numerical T-matrix solution for polarized scattering from a cluster of spatially oriented, nonspherical scatterers,” Q. Appl. Math. 12(3), 154–158 (1996).
[Crossref]

Iati, M. A.

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

Jin, Y.-Q.

Y.-Q. Jin and X. Huang, “Numerical T-matrix solution for polarized scattering from a cluster of spatially oriented, nonspherical scatterers,” Q. Appl. Math. 12(3), 154–158 (1996).
[Crossref]

Khlebtsov, N. G.

N. G. Khlebtsov, “T-matrix method in plasmonics: An overview,” J. Quant. Spectrosc. Radiat. Transfer 123, 184–217 (2013).
[Crossref]

Kocifaj, M.

G. Videen and M. Kocifaj, Optics of cosmic dust, vol. 79 (Springer Science & Business Media, 2002).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

Lafait, J.

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple-scattering problems,” J. Mod. Opt. 49(13), 2129–2152 (2002).
[Crossref]

Le Ru, E.

W. Somerville, B. Auguié, and E. Le Ru, “SMARTIES: User-friendly codes for fast and accurate calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 174, 39–55 (2016).
[Crossref]

Le Ru, E. C.

M. R. A. Majić, B. Auguié, and E. C. Le Ru, “Spheroidal harmonic expansions for the solution of Laplace’s equation for a point source near a sphere,” Phys. Rev. E 95(3), 033307 (2017).
[Crossref]

B. Auguié, W. R. C. Somerville, S. Roache, and E. C. Le Ru, “Numerical investigation of the Rayleigh hypothesis for electromagnetic scattering by a particle,” J. Opt. 18(7), 075007 (2016).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Accurate and convergent T-matrix calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 160, 29–35 (2015).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “A new numerically stable implementation of the T-matrix method for electromagnetic scattering by spheroidal particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 153–168 (2013).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Severe loss of precision in calculations of T-matrix integrals,” J. Quant. Spectrosc. Radiat. Transfer 113(7), 524–535 (2012).
[Crossref]

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

J. Grand and E. C. Le Ru, “Practical implementation of accurate finite-element calculations for electromagnetic scattering by nanoparticles,” Plasmonics (2019).

E. C. Le Ru and P. G. Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy: and related plasmonic effects (Elsevier, 2009).

Lemmer, U.

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

lin Xu, Y.

Liu, L.

L. Liu, M. I. Mishchenko, and W. P. Arnott, “A study of radiative properties of fractal soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109(15), 2656–2663 (2008).
[Crossref]

Loke, V. L.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58(5-6), 528–544 (2011).
[Crossref]

Luong, J. H. T.

Z. Zhong, S. Patskovskyy, P. Bouvrette, J. H. T. Luong, and A. Gedanken, “The surface chemistry of au colloids and their interactions with functional amino acids,” J. Phys. Chem. B 108(13), 4046–4052 (2004).
[Crossref]

Ma, X.

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transfer 55(5), 535–575 (1996).
[Crossref]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996).
[Crossref]

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11(11), 2851–2861 (1994).
[Crossref]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London, Ser. A 433(1889), 599–614 (1991).
[Crossref]

Majic, M. R. A.

M. R. A. Majić, B. Auguié, and E. C. Le Ru, “Spheroidal harmonic expansions for the solution of Laplace’s equation for a point source near a sphere,” Phys. Rev. E 95(3), 033307 (2017).
[Crossref]

Markkanen, J.

J. Markkanen and A. J. Yuffa, “Fast superposition T-matrix solution for clusters with arbitrarily-shaped constituent particles,” J. Quant. Spectrosc. Radiat. Transfer 189, 181–188 (2017).
[Crossref]

Martin, T.

T. Martin, “T-matrix method for closely adjacent obstacles,” J. Quant. Spectrosc. Radiat. Transfer 234, 40–46 (2019).
[Crossref]

Maystre, D.

D. Maystre and M. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26(9), 2201–2204 (1985).
[Crossref]

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Millar, R.

R. Millar, “Rayleigh hypothesis in scattering problems,” Electron. Lett. 5(17), 416–417 (1969).
[Crossref]

Mishchenko, M. I.

L. Liu, M. I. Mishchenko, and W. P. Arnott, “A study of radiative properties of fractal soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109(15), 2656–2663 (2008).
[Crossref]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transfer 55(5), 535–575 (1996).
[Crossref]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996).
[Crossref]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

Nieminen, T. A.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58(5-6), 528–544 (2011).
[Crossref]

Patskovskyy, S.

Z. Zhong, S. Patskovskyy, P. Bouvrette, J. H. T. Luong, and A. Gedanken, “The surface chemistry of au colloids and their interactions with functional amino acids,” J. Phys. Chem. B 108(13), 4046–4052 (2004).
[Crossref]

Peterson, B.

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8(10), 3661–3678 (1973).
[Crossref]

Radney, J. G.

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

Raschke, M. B.

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

Roache, S.

B. Auguié, W. R. C. Somerville, S. Roache, and E. C. Le Ru, “Numerical investigation of the Rayleigh hypothesis for electromagnetic scattering by a particle,” J. Opt. 18(7), 075007 (2016).
[Crossref]

Rother, T.

T. Rother, Electromagnetic wave scattering on nonspherical particles (Springer, 2009).

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58(5-6), 528–544 (2011).
[Crossref]

Saija, R.

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

Somerville, W.

W. Somerville, B. Auguié, and E. Le Ru, “SMARTIES: User-friendly codes for fast and accurate calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 174, 39–55 (2016).
[Crossref]

Somerville, W. R. C.

B. Auguié, W. R. C. Somerville, S. Roache, and E. C. Le Ru, “Numerical investigation of the Rayleigh hypothesis for electromagnetic scattering by a particle,” J. Opt. 18(7), 075007 (2016).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Accurate and convergent T-matrix calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 160, 29–35 (2015).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “A new numerically stable implementation of the T-matrix method for electromagnetic scattering by spheroidal particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 153–168 (2013).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Severe loss of precision in calculations of T-matrix integrals,” J. Quant. Spectrosc. Radiat. Transfer 113(7), 524–535 (2012).
[Crossref]

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15–24 (1961).
[Crossref]

Stilgoe, A. B.

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58(5-6), 528–544 (2011).
[Crossref]

Stout, B.

B. Stout, J. C. Auger, and A. Devilez, “Recursive T matrix algorithm for resonant multiple scattering: applications to localized plasmon excitations,” J. Opt. Soc. Am. A 25(10), 2549–2557 (2008).
[Crossref]

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple-scattering problems,” J. Mod. Opt. 49(13), 2129–2152 (2002).
[Crossref]

Ström, S.

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8(10), 3661–3678 (1973).
[Crossref]

Subramaniam, V.

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

Sundheimer, M. L.

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

Theobald, D.

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

Tian, L.

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transfer 55(5), 535–575 (1996).
[Crossref]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

Tucker, E.

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

van Amerongen, H.

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

Videen, G.

G. Videen and M. Kocifaj, Optics of cosmic dust, vol. 79 (Springer Science & Business Media, 2002).

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3(4), 825–839 (1971).
[Crossref]

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45(6), 1417–1429 (1969).
[Crossref]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

Wriedt, T.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Yang, H. U.

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

Yang, P.

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 138, 17–35 (2014).
[Crossref]

Yeh, C.

You, R.

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

Yuffa, A. J.

J. Markkanen and A. J. Yuffa, “Fast superposition T-matrix solution for clusters with arbitrarily-shaped constituent particles,” J. Quant. Spectrosc. Radiat. Transfer 189, 181–188 (2017).
[Crossref]

Zachariah, M. R.

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

Zangmeister, C. D.

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

Zhong, Z.

Z. Zhong, S. Patskovskyy, P. Bouvrette, J. H. T. Luong, and A. Gedanken, “The surface chemistry of au colloids and their interactions with functional amino acids,” J. Phys. Chem. B 108(13), 4046–4052 (2004).
[Crossref]

ACM Trans. Math. Softw. (1)

D. Amos, “Algorithm 644, a portable package for Bessel functions of a complex argument and nonnegative order,” ACM Trans. Math. Softw. 12(3), 265–273 (1986).
[Crossref]

Ann. Phys. (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Appl. Opt. (1)

Astrophys. J. (1)

R. Saija, M. A. Iati, F. Borghese, P. Denti, S. Aiello, and C. Cecchi-Pestellini, “Beyond Mie theory: The transition matrix approach in interstellar dust modeling,” Astrophys. J. 559(2), 993–1004 (2001).
[Crossref]

Electron. Lett. (1)

R. Millar, “Rayleigh hypothesis in scattering problems,” Electron. Lett. 5(17), 416–417 (1969).
[Crossref]

Environ. Sci. Technol. (1)

J. G. Radney, R. You, X. Ma, J. M. Conny, M. R. Zachariah, J. T. Hodges, and C. D. Zangmeister, “Dependence of soot optical properties on particle morphology: Measurements and model comparisons,” Environ. Sci. Technol. 48(6), 3169–3176 (2014).
[Crossref]

J. Acoust. Soc. Am. (1)

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45(6), 1417–1429 (1969).
[Crossref]

J. Math. Phys. (1)

D. Maystre and M. Cadilhac, “Singularities of the continuation of fields and validity of Rayleigh’s hypothesis,” J. Math. Phys. 26(9), 2201–2204 (1985).
[Crossref]

J. Mod. Opt. (2)

T. A. Nieminen, V. L. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix method for modelling optical tweezers,” J. Mod. Opt. 58(5-6), 528–544 (2011).
[Crossref]

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple-scattering problems,” J. Mod. Opt. 49(13), 2129–2152 (2002).
[Crossref]

J. Opt. (1)

B. Auguié, W. R. C. Somerville, S. Roache, and E. C. Le Ru, “Numerical investigation of the Rayleigh hypothesis for electromagnetic scattering by a particle,” J. Opt. 18(7), 075007 (2016).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Phys. Chem. B (1)

Z. Zhong, S. Patskovskyy, P. Bouvrette, J. H. T. Luong, and A. Gedanken, “The surface chemistry of au colloids and their interactions with functional amino acids,” J. Phys. Chem. B 108(13), 4046–4052 (2004).
[Crossref]

J. Phys. Chem. C (1)

R. Gill, L. Tian, W. R. C. Somerville, E. C. Le Ru, H. van Amerongen, and V. Subramaniam, “Silver nanoparticle aggregates as highly efficient plasmonic antennas for fluorescence enhancement,” J. Phys. Chem. C 116(31), 16687–16693 (2012).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (10)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: A review,” J. Quant. Spectrosc. Radiat. Transfer 55(5), 535–575 (1996).
[Crossref]

J. Markkanen and A. J. Yuffa, “Fast superposition T-matrix solution for clusters with arbitrarily-shaped constituent particles,” J. Quant. Spectrosc. Radiat. Transfer 189, 181–188 (2017).
[Crossref]

T. Martin, “T-matrix method for closely adjacent obstacles,” J. Quant. Spectrosc. Radiat. Transfer 234, 40–46 (2019).
[Crossref]

N. G. Khlebtsov, “T-matrix method in plasmonics: An overview,” J. Quant. Spectrosc. Radiat. Transfer 123, 184–217 (2013).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Accurate and convergent T-matrix calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 160, 29–35 (2015).
[Crossref]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 138, 17–35 (2014).
[Crossref]

L. Liu, M. I. Mishchenko, and W. P. Arnott, “A study of radiative properties of fractal soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109(15), 2656–2663 (2008).
[Crossref]

W. Somerville, B. Auguié, and E. Le Ru, “SMARTIES: User-friendly codes for fast and accurate calculations of light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer 174, 39–55 (2016).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Severe loss of precision in calculations of T-matrix integrals,” J. Quant. Spectrosc. Radiat. Transfer 113(7), 524–535 (2012).
[Crossref]

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “A new numerically stable implementation of the T-matrix method for electromagnetic scattering by spheroidal particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 153–168 (2013).
[Crossref]

Opt. Spectrosc. (1)

V. G. Farafonov, “Applicability of the T-matrix method and its modifications,” Opt. Spectrosc. 92(5), 748–760 (2002).
[Crossref]

Phys. Rev. A (1)

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

Phys. Rev. B (2)

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
[Crossref]

H. U. Yang, J. D’Archangel, M. L. Sundheimer, E. Tucker, G. D. Boreman, and M. B. Raschke, “Optical dielectric function of silver,” Phys. Rev. B 91(23), 235137 (2015).
[Crossref]

Phys. Rev. D (2)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3(4), 825–839 (1971).
[Crossref]

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8(10), 3661–3678 (1973).
[Crossref]

Phys. Rev. E (1)

M. R. A. Majić, B. Auguié, and E. C. Le Ru, “Spheroidal harmonic expansions for the solution of Laplace’s equation for a point source near a sphere,” Phys. Rev. E 95(3), 033307 (2017).
[Crossref]

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London, Ser. A 433(1889), 599–614 (1991).
[Crossref]

Q. Appl. Math. (3)

Y.-Q. Jin and X. Huang, “Numerical T-matrix solution for polarized scattering from a cluster of spatially oriented, nonspherical scatterers,” Q. Appl. Math. 12(3), 154–158 (1996).
[Crossref]

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15–24 (1961).
[Crossref]

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20(1), 33–40 (1962).
[Crossref]

Other (7)

E. C. Le Ru and P. G. Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy: and related plasmonic effects (Elsevier, 2009).

J. Grand and E. C. Le Ru, “Practical implementation of accurate finite-element calculations for electromagnetic scattering by nanoparticles,” Plasmonics (2019).

T. Rother, Electromagnetic wave scattering on nonspherical particles (Springer, 2009).

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

G. Videen and M. Kocifaj, Optics of cosmic dust, vol. 79 (Springer Science & Business Media, 2002).

W. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press series on electromagnetic waves (Van Nostrand Reinhold, 1990).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

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Figures (5)

Fig. 1.
Fig. 1. (a) Prolate spheroid with major semi-axis $c$, minor semi-axis $a$, and focal distance $f = \sqrt {c^{2}-a^{2}}$. Dashed black and red circumferences represent the circumscribed (CS) and focal sphere (FS) of radius $c$ and $f$, respectively. Digits $1$, $2$, and $3$ demarcate the far, near, and interior zones. (b) Two spheroids and three reference frames with aligned orientation but different origins: $O$, $O_{i}$, and $O_{j}$. (c–f) Clusters of two identical spheroids side by side, with $c$ varied to increase the degree of overlap: (c) CSs intersect but not FSs, (d) FSs intersect, (e) FSs intersect with other spheroid, (f) $r_{ji}<c$. In the latter cases, the solid red circumference represents the transformed singularity of irregular VSWFs when translated by a vector $\mathbf {r}_{ji}$.
Fig. 2.
Fig. 2. Spectra of the scattering and absorption cross-sections (normalised by the geometrical cross-section $\sigma _\textrm {geo} = 2\pi a^{2}$) for the dimers in Figs. 1(c)–1(f). Six $c$-values are considered with fixed $a = 20$ nm and gap of $10$ nm. Finite-element calculations (black circles) are compared with $T$-matrix data for $n_\textrm {c} = 20$ (solid red line). The reference $T$-matrix data (dashed blue line) is for two spheroids isolated from each other (non-interacting).
Fig. 3.
Fig. 3. Top: Spectra of normalised extinction cross-section for the $c=52$ nm case in Fig. 2, calculated using terms with different $n_\textrm {c}$ values and compared with finite-element (FE) calculations (black dots). Bottom: The FE value (thick line) at specified wavelength compared to terms calculations in double (dashed line) and quad (solid line) precision for increasing $n_\textrm {c}$.
Fig. 4.
Fig. 4. (a) Local-field enhancement factor $I/I_{0}$ and (b) magnitude of the error relative to FE results mapped along three symmetry planes for the $c=36$ nm case in Fig. 2 at $\lambda = 436$ nm, calculated using terms in double precision with $n_\textrm {c} = 40$. White regions outside the spheroids but inside their focal spheres indicate divergence. (c) Line scan of the enhancement factor along $\mathbf {k}$ and through the point of intersection of all three symmetry planes. Calculations with different values of $n_\textrm {c}$ show convergence to the FE results, all plotted on linear and logarithmic scales for clarity. (d) Analogous line scan for the dimer with $c=44$ nm, calculated using quad precision and showing slower convergence in the overlap region.
Fig. 5.
Fig. 5. Top: normalised extinction spectra for a silver sphere of radius $r=15$ nm near a silver spheroid of $a=20$ and $c=60$ nm, with $\sigma _\textrm {geo} = \pi (a^{2}+r^{2})$. The geometry has two symmetry planes and $5$ nm gap ($r_{ij}=40$ nm). Bottom: maps of the local-field intensity enhancement (calculated using terms in quadruple precision with $n_\textrm {c} = 50$) and the error relative to FE calculations for $\lambda =430$ nm.

Equations (5)

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f ( r ) = l = 1 a ~ l w ~ l ( r ) f inc ( r ) + l = 1 a l w l ( r ) f sca ( r ) = W ~ ( r ) a ~ + W ( r ) a ,
a = T a ~ ,
σ ext = R { a ~ a } k med 2 and σ sca = a a k med 2 ,
f sca ( r ) = i = 1 N f sca ( i ) ( r r i ) = i = 1 N W ( r r i ) c i ( i ) ,
c i ( i ) = T i ( a ~ ( i ) + j i O ( i , j ) c j ( j ) c ~ j ( i ) ) e ~ i ( i ) ,